pcf: Pair Correlation Function

Description

Estimate the pair correlation function.

Usage

pcf(X, ...)

Arguments

Either the observed data point pattern, or an estimate of its $K$ function, or an array of multitype $K$ functions (see Details).

...

Other arguments passed to the appropriate method.

Value

Either a function value table (object of class "fv", see fv.object) representing a pair correlation function, or a function array (object of class "fasp", see fasp.object) representing an array of pair correlation functions.

Details

The pair correlation function of a stationary point process is

g (r) = \frac{K^{'} (r)}{2 π r}

where $K'(r)$ is the derivative of $K(r)$, the reduced second moment function (aka ``Ripley's $K$ function'') of the point process. See Kest for information about $K(r)$. For a stationary Poisson process, the pair correlation function is identically equal to 1. Values $g(r) < 1$ suggest inhibition between points; values greater than 1 suggest clustering.

We also apply the same definition to other variants of the classical $K$ function, such as the multitype $K$ functions (see Kcross, Kdot) and the inhomogeneous $K$ function (see Kinhom). For all these variants, the benchmark value of $K(r) = \pi r^2$ corresponds to $g(r) = 1$.

This routine computes an estimate of $g(r)$ either directly from a point pattern, or indirectly from an estimate of $K(r)$ or one of its variants.

This function is generic, with methods for the classes "ppp", "fv" and "fasp".

If X is a point pattern (object of class "ppp") then the pair correlation function is estimated using a traditional kernel smoothing method (Stoyan and Stoyan, 1994). See pcf.ppp for details.

If X is a function value table (object of class "fv"), then it is assumed to contain estimates of the $K$ function or one of its variants (typically obtained from Kest or Kinhom). This routine computes an estimate of $g(r)$ using smoothing splines to approximate the derivative. See pcf.fv for details.

If X is a function value array (object of class "fasp"), then it is assumed to contain estimates of several $K$ functions (typically obtained from Kmulti or alltypes). This routine computes an estimate of $g(r)$ for each cell in the array, using smoothing splines to approximate the derivatives. See pcf.fasp for details.

References

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

Examples

Run this code

# ppp object
  data(simdat)
  <testonly>simdat <- simdat[seq(1,simdat$n, by=4)]</testonly>
  p <- pcf(simdat)
  plot(p)

  # fv object
  K <- Kest(simdat)
  p2 <- pcf(K)
  plot(p2)

  # multitype pattern; fasp object
  data(betacells)
  <testonly>betacells <- betacells[seq(1,betacells$n, by=10)]</testonly>
  p <- pcf(alltypes(betacells, "K"))
  plot(p)

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